Counting Frabjous Numbers
A 'frabjous' number is a 3-digit number where all digits are odd, and no adjacent digits are the same.
The odd digits are: 1, 3, 5, 7, 9. There are 5 odd digits available.
We need to determine the number of choices for each of the three positions (hundreds, tens, units).
- Hundreds digit: This digit must be odd. There are 5 choices (1, 3, 5, 7, 9).
- Tens digit: This digit must also be odd, but it cannot be the same as the hundreds digit. Since one odd digit is used for the hundreds place, there are $5 - 1 = 4$ choices remaining.
- Units digit: This digit must be odd and cannot be the same as the tens digit. It can be the same as the hundreds digit. Since one odd digit is used for the tens place, there are $5 - 1 = 4$ choices remaining.
Calculating Total Frabjous Numbers
To find the total number of possible frabjous numbers, we multiply the number of choices for each digit:
Total Numbers = (Choices for hundreds digit) $\times$ (Choices for tens digit) $\times$ (Choices for units digit)
Total Numbers = $5 \times 4 \times 4$
Total Numbers = $80$
Therefore, there are 80 such frabjous numbers.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.
